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192       XIII    INTEGRATION

By writing each /*,- as a linear combination of four positive measures
(13.3.6), we may assume that the measures \JLJ are all positive. Then we take

r

^ == X Af/> and apply the result of (13.15.1) to each \JLJ g L

(13.15.3) (i) With respect to the order relation on MR(X), any two real
measures \i, v have a least upper bound sup(/j, v) and a greatest lower bound
inf(/*, v). For each real measure \JL put /z+ = sup^u, 0) and /i~ = sup( — /*, 0);
then we have

(13.15.3.1)   inf(^+, iT) = 0,      /* =
and, for any two real measures /i, v,

v = sup(/z, v) + inf(^, v),
(13.15.3.2)

i, v) = i(/^ + v + b — vl)>
inf(//, v) = i(/( + v — |/i — v|).

(ii)   Le^ 1 be a positive measure and let g^ g2 be real-valued locally
k-integrable functions on X. If fa = #1 * A <2/?Ģ/ ^J2 = ^2 • /



m particular, for any real-valued locally X-integrable function g,

(13.15.3.4)               (g • A)+ =g+ - ^       (g • A)' = <T ' A.

PFe /?m;e ^ • A ^ g2 ' A if and only if gą(x) ^ g2(x) almost everywhere with
respect to L

(iii) Let (gn) be an increasing sequence of locally A-integrable real-valued
functions. Then the increasing sequence of measures gn • 1 is bounded above in
MR(X) if and only if the function sup gn is locally k-integrable, and in that case
we have                                       n

(13.15.3.5)                        supte.-A

To prove (i), we remark that the measures n, v can be written in the form
^ = ^-^2, v = v1-v2, where /^, /*2, Y! and v2 are positive (13.3.6).
Applying (13.15.2) to the four measures jjLl9 /i2, v1? v2 we see (by using
(13.13.2)) that /i = u - 1 and v = i? • A, where A is a positive measure and u, v        m   r